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Article Dans Une Revue Geometric And Functional Analysis Année : 2020

Branching geodesics in sub-Riemannian geometry

Thomas Mietton
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Luca Rizzi

Résumé

In this note, we show that sub-Riemannian manifolds can contain branching normal minimizing geodesics. This phenomenon occurs if and only if a normal geodesic has a discontinuity in its rank at a non-zero time, which in particular for a strictly normal geodesic means that it contains a non-trivial abnormal subsegment. The simplest example is obtained by gluing the three-dimensional Martinet flat structure with the Heisenberg group in a suitable way. We then use this example to construct more general types of branching.
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Dates et versions

hal-02493682 , version 1 (28-02-2020)
hal-02493682 , version 2 (07-06-2020)

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Thomas Mietton, Luca Rizzi. Branching geodesics in sub-Riemannian geometry. Geometric And Functional Analysis, 2020, 30, pp.1139 - 1151. ⟨10.1007/s00039-020-00539-z⟩. ⟨hal-02493682v2⟩
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